Is it not wonderful that they should have failed to deduce from the works of God the vitally momentous consideration that a perfect consistency can be nothing but an absolute truth?

Edgar Allan Poe, quoted by Thomas Bertonneau.

Our religion is under attack. We are under attack. To defeat our opponents we must first convince ourselves that our doctrines are true and our ways are good. How can we know these things?

To know despite uncertainty and opposition we must have an argument. Not “argument” in the sense of fighting, but having a persuasive case. We must persuade ourselves first and then (God willing) we can persuade others.

My emphasis here is on persuading ourselves. If we know how we know, and we know our doctrine is true, only then can we defend ourselves.

Using arguments to defend one’s doctrine goes by the name “apologetics,” especially in Christendom. There has always been disagreement over the best type of apologetics. A recent development (about a hundred years old) is the school of Presuppositional Apologetics. It arose within Protestantism of the Reformed variety. Like all important terms, “presuppositional apologetics” has many layers of meaning. There are a few root ideas and many subsequent developments. As with all schools of thought, the presuppositional schoolmen sometimes go off into the weeds.

But the root idea of presuppositionalism is decisive for apologetics: All thinking is governed by presuppositions, meaning basic beliefs that are accepted without proof and often without conscious awareness.

(Thinking is also governed by other factors: emotions, sensations, memories, etc. This work deals with the objective content of thought, that is, thinking about matters that can be true or false.)

The atheist, for example, says there is no evidence for God and therefore he does not believe. There *is* evidence, but the atheist presupposes that reality is material only and that to be rational is to think materialistically. He does not examine and prove these notions, he accepts them as his presuppositions. Judging the evidence by his presupposition of atheism, the atheist sees only atheism and is surprised that anyone could doubt what he finds obvious.

Likewise the socio-political liberal presupposes (*inter alia*) that a toxic wh*t* s*pr*m*cy suffuses America like smog, that all people are naturally equal, and that individual persons define their identities in any way they wish. No modern liberal tests these ideas; they are presuppositions.

The vast majority of mankind has always lived by accepting the presuppositions of their community. But we live in volatile times. The old presuppositions of America are in wide disrepute and other presuppositions are competing for customers. Many consumers are confused by the variety of products. How can the shopper discern the best product?

*

Presuppositions are not proved because proof must always start somewhere. If everything must be proved there is an endless regression of proofs, and we know nothing. If “proof” must be done formally, by calling on other truths as witnesses, then there must be beliefs that are not proved. These unproved beliefs are presuppositions.

And yet…

Even though a presupposition is not proved, we must have evidence to support it. We want our presuppositions to be true, so we do not choose them randomly. At the very least, we have some outcome we *want* to be true and we look around for presuppositions or evidence to support it, which is the way many people seem to operate.

So presuppositions must be tested. The verb “test” is related to “prove,” but whereas proof is more formally defined within the discipline of epistemology, “testing” is less formal. It is more intuitive.

Intuition is the faculty, or the process, of knowing something immediately, without going through a process of formal reasoning. Although God is the ultimate basis of all our knowing because He is the basis of all truth, yet within ourselves, intuition is the ultimate basis of all our knowledge. Truth does not help us until we grasp it, and intuition is the ultimate means by which we grasp truth.

Presuppositions, the basis of all our formal knowledge, are by definition not proved. But we cannot choose our presuppositions randomly, because we have an intuitive sense that some things really are true or false. *So how do we test our presuppositions?*

This is *the* key question. We can know truth only by identifying our presuppositions and then testing them. To know the truth about anything we must first have the right way of thinking about it, so first we must know that we have the right way of thinking.

So how do we test presuppositions?

In two ways. True presuppositions must accord with our intuitive sense of what things are true and false. And the presuppositions, along with the other truths that they prove, must be consistent. Truth is a consistent system.

So we must have a consistent system that accords with our intuitive sense of how the world operates, but with two major *caveats* which I discuss shortly.

*

My model here is Euclidian Geometry. In his celebrated *Elements*, Euclid attempted to show how all geometric knowledge could be derived from a minimal set of axioms which express primitive, apparently-obvious facts about points, lines and planes. But system-building sometimes leads to unexpected results. When Euclid attempted to prove his Parallel Theorem (“Through any given point not on a line there passes exactly one line parallel to that line in the same plane”) he found that his system of axioms could not prove it. Instead, he had to adopt the Parallel *Postulate*, and express this truth as an unproved, but intuitively obvious, idea.

In Euclid’s system, presuppositions (axioms) lead to other propositions (theorems) that *are* proved, and the entire system is then judged as a whole. If no contradiction can be derived from the presuppositions, and if both the presuppositions and the other propositions they support appear true to our intuition (some theorems require ingenuity to be rendered self-evident), then we have confidence that the system is true.

Other systems of thought have similarity to Euclidian geometry. In other fields, the definitions may not be so sharp, the axioms may not be universally agreed-upon, and the reasoning may not be as precise, but in *form* the systems are similar: All is based on a set of presuppositions, which are known by intuition. Man, as a rational animal, cannot think otherwise.

*

I said that we know we have truth when we have a consistent system that accords with our intuitive sense of how the world operates, but with two major *caveats*. Here they are.

The first is this: God is the ultimate determiner of reality, and therefore of truth. God equipped man with the ability to grasp many truths about the world without consciously thinking of God. But there are truths that mankind cannot know unless God reveals them to him in written form. This written Revelation is the Bible.

The second *caveat *is this: Man’s intuition can be perverted, and therefore he must be willing to test his beliefs. Our intuitions are sometimes mistaken.

*

All people have presuppositions which control the outcome of their thinking. Therefore if we are to have integrity we must test our presuppositions. They seem right to us, or else we would not have presupposed them. But perhaps they are not consistent. Since my intuition tells me that truth exists and is self-consistent, I conclude that an inconsistent system of presuppositions is false at some point, and must be modified.

We must also test our presuppositions by their consequences. For example, the classic error of hard materialism runs as follows:

Major premise: If hard materialism is true, then consciousness is not real.

Minor premise: Hard materialism is true.

Conclusion: Consciousness is not real.

But the correct reasoning is as follows:

Major premise: If hard materialism is true, then consciousness is not real.

Minor premise: Consciousness is real. [We know this intuitively]

Conclusion: Hard materialism is false.

We correct presuppositions which lead to false conclusions.

*

Authority also plays a role in epistemology. Some things can only be known because a trustworthy authority tells us. Most of us cannot know the latest facts from astrophysics unless an astrophysicist tells us, and he speaks the truth, and we understand and believe him. *Theoretically*, some people could take the time to learn enough astrophysics to understand the latest research. But the vast majority of mankind lacks the intelligence, spare time, and determination to be able to follow astrophysics on their own. For all practical purposes, the rest of mankind must trust the authorities.

We also cannot know that we are in danger of Hell, and that Jesus Christ is our only rescue, unless God tells us this truth and we believe Him. We cannot derive these truths from our own resources.

This relates to the question “How can we know Christianity is true?”

To answer the question properly we must understand that any serious system of thought has a highest authority. The atheist’s highest authority is mankind: Either man the individual (“I make my own truth”) or man the group (“Each culture defines its own truth.”)

So the contention is *not* between the rationalists who accept the findings of modern science versus those crazy fundamentalists who rely on the Bible. It is between those who see mankind as the ultimate determiner of truth versus those who acknowledge God as the Supreme Being. And the atheist, if he is to have integrity, cannot simply presuppose that atheism is correct and that mankind is the ultimate determiner of truth. To have integrity, a man must test his system.

The ultimate reason Christianity is true is this: As a system, Christianity works better than any other system. It is consistent within itself and it best explains the facts of the world. The other systems work less well, or poorly. This is the key insight of presuppositional apologetics.

When the role of presuppositions is understood, the *non-presuppositional* apologetics goes to work. Evidential apologetics looks at the evidence and shows how it supports and is consistent with Christianity.

A cynic would point to all the different versions of Christianity. But as a defeater of Christianity this objection fails. All the non- and anti-Christian systems also have different versions. And there is a ready explanation for the phenomenon pointed out by the cynic: One of the versions of Christianity is the optimal one.

*

So the person who argues against Christianity cannot just assume his system is correct. It’s not enough to point out apparent problems with Christianity. Within the atheist’s system, Christianity appears false, but within the system of Christianity, atheism is false. So whose system is better? Accusing the biblical God of being a tyrant, or accusing the biblical authors of lying, or pooh-poohing accounts of the Resurrection fail radically as disproof of Christianity. When two systems contend, a deeper level of thinking is required.

The basic test of systems is to ask, **Which system works better?**

The discernment of the answer can be made in different ways, at different levels of sophistication or complexity, but when the answer comes to you, must be grasped intuitively. Christian doctrine expresses this by saying that faith – – the ability to believe in Jesus Christ as God and Savior – – is a gift that God gives only to some. The evidence can be accumulated, but some people respond correctly and some do not.

What is some of this evidence for Christianity? The evidence fills libraries so it is only appropriate here to give an outline. This essay is about the big picture, not the details.

Atheistic governments unchained from the traditional Christianity of Western man have committed mass atrocities, not incidentally in moments of passion, but as expressions of their essence. Atheistic philosophy becomes incoherent by making human understanding – – which constantly changes – – the ultimate ground of truth. The social systems of the modern world – – which shake their fists at God – – are spiraling out of control.

The system of contemporary Western atheism can also be criticized intellectually. This is a more esoteric and elitist critique that is not for everyone. But if the world does not begin – – either temporally or logically or metaphysically – -with mind, love and purpose already existent, these can never come into being. Since they *are* here, they must have come from God, the necessary eternal Being.

And what is this Being like? As noted above, the God of the Bible is characterized by mind, love and purpose. He has the power and wisdom to create an ordered universe, and to create mankind capable of detecting this order. If such a Being exists, He is capable of communicating with us, and of validating His message. Since even contemporary science knows that matter is not eternal, this God was able to create matter from nothing other than Himself. Being this powerful, God could have planned and executed the campaign of salvation that is described in the Bible, and He could have intervened with miracles to keep His campaign advancing, and to supply proof of His intentions when necessary. (Especially the miracles of the Exodus and of the life and resurrection of Jesus Christ.) If God exists, there is nothing foolish about the Bible.

If the Christian system is true, man is a fallen but noble creature. He can regain his former glory by acknowledging and submitting to God his maker through faith in Jesus Christ, his Savior. If Christianity is true, the cosmos makes sense because it was created by a Being who is Sense itself. If Christianity is true, man can know how to honor his God and to live in peace with his fellow man. If Christianity is true, mankind does not need to struggle to “change the world,” or to “make a difference,” or to “be the change that you want to see.” When he is in Christ, man has all the righteousness he needs.

That is why Christians call on all mankind to have faith in Jesus Christ, the Son of God, and be saved.

*

And what about defending America? The basic point which the patriot should make to the liberal, the leftist, the gl0bαlιst, the 0lιgαrch, the deep-stαtεr, the Alphabet activist, et al, is this:

* Your system controls the world. Your system does not work*.

Consider the world under your rule: Anτιfα violence, C0vιd tyrαηηy and socio-economic disruption, officially-endorsed rαcιαl hαtrεd, vengeful and tyrαnnιcαl government, filthy popular culture, debased standards leading to increasing incompetence, hostility between men and women and between parents and children, etc.

Your system controls all Western governments. Like communists blaming counterrevolutionaries, your side blames “Τrυmρ supporters.” But your system has ruled for decades. Your system caused the current mess.

Your system does not work.

Certain things have to be presupposed because they are obvious and don’t actually need proof, three things really, i.e. that God exists, created the world, and expects us to live morally. Anyone who denies these things is trash and not worth dealing with to be honest. Even if you “converted” them you would only be converting a trojan horse that will spread degeneracy in your churches. Where the Calvinists go wrong in presuppositional apologetics, however, is in presupposing their entire denominc system of predestination and so on. If you are gonna engage in presuppositional apologetics, the number of presuppositions must be small, and they must be the obvious ones.

It’s about comparing systems by identifying their presuppositions and then subjecting the systems to an honest test.

“The atheist, for example, says there is no evidence for God and therefore he does not believe. There is evidence, but the atheist presupposes that reality is material only and that to be rational is to think materialistically.”

I disagree. But maybe that is my 4th presupposition. Anyone who denies my main 3 presuppositions (the existence of God, that God created the world, that God wants us to live morally) is simply evil and denies those 3 presuppositions out of a desire to defend their evil ways and is not honest in their claims to disbelieve these 3 rather obvious and unquestionable presuppositions. I will not argue with them as if they honestly presuppose a merely material world, because I do not believe they do. I think pretending atheists are honest in their claim to not believe in these 3 presuppositions is the problem everyone in apologetics has been making for 200 years or so and if your brand of so-called “presuppositional apologetics” will continue that mistake then it is of no value. So presupposition 5, those who defend sin lie about their presuppositions. Therefore analizing their supposed presuppositions is a waste and instead you should simply build on your own. Unless the opponent also acknowledges the main 3 presuppositions, in which case where theu disagree with later presuppositions we can analyze theirs versus ours. But even analyzing the lying “presuppositions” of those who claim to deny the first 3 is a waste because they are lying.

I said that this article is about the rational content of thought. But there is also a lot of irrationality in mankind. If your interlocutor is a swine, do not cast your pearls.

Oh my gosh, thanks, Alan. Presuppositional apologetics is something I’ve been meaning to dig into for years, on account of the fact that (primarily due to what I’ve written about Anselm) I’ve been characterized as a presuppositionalist myself. Is there a particular book you would recommend as a point of entry to the topic? Van Til, or one of his commentators?

Several things occurred to me as I read:

I never realized this before, because I didn’t remember the Parallel Postulate from high school geometry (which I *totally adored,* it was intellectual catnip), but it appears that, being one of the *obviously true* (and entirely consistent) truths that Euclidean geometry is capable of expressing but cannot prove, *the Parallel Postulate is thus a demonstration of the truth of Gödel’s Incompleteness Theorem.* You think? Wow. If so, this will be my go to example of such a truth from now on, because so many people are likely to get it.

Second, the connection of presuppositionalism to Pragmatism, which I had never before noticed: the propositional system derived from the presuppositions must *work.* It must agree with experience, and one way we can tell whether it does so agree is to see whether it leads to adverse results when carried into practice or experiment (this is the Pragmatic test (I just realized it is also the Gedanken Policy Test)). A consistent system might have nothing to do with our world, after all; if your consistent system suggests that you can breathe water, e.g., you’ve got a problem. Pragmatism has the popular reputation of being all loosey goosey with its criteria of truth, but in fact – when we treat it through the lens of the test of presuppositional systems you here propose – it turns out to be crushingly rigorous, rigorous to a fault – indeed, exactly that.

Peirce would have been pleased with this; Bacon, too. And Popper.

Third, and of most interest: on Gödel, like any other sort of logical calculus, a consistent theological calculus *cannot be completed.* There will *always* be truths that it can express, but that it cannot demonstrate.

Now, this has several delicious consequences.

First, the mystics and saints have always and universally insisted – not just within the Church, but in all religious traditions – that cataphatic theology – propositional systems couched in defined terms – cannot possibly adequate to Reality. They must – logically must – be right!

Second, this means that the development of doctrine *must be endless.* This is just another way of saying that it cannot be possible to finish a consistent account of the Infinite One (Gödel again); it cannot be possible for us to finish understanding God.

When the notion is expressed in this way, it is immediately obvious that to think otherwise is just absurd, no?

And that casts all theology into a humbler light. We’ve only been scratching the surface, so far. As Lewis remarked, we must remember – always, unto ages of ages – that we are still only the Early Christians. Likewise must we remember, always, that we are still only the first few scientists or philosophers.

Ten thousand years from now, we all of the last millennium will appear as early developments from Heraclitus, Moses, Philo, Paul, Augustine, & alii. Footnotes to Plato, all of us, as Whitehead rightly remarked.

Think then of our puny doctrinal attempts upon the Most High as a sort of Pelagianism; as a sort of Babelonian pride. It is a humiliation, is it not? And most salutary. The Orthodox hesychasts, the Taoists, and the Zen Buddhists – and St. John of the Cross, and The Cloude of Unknowyng – make a very good point: Be still, then, and know that I am God; Let all mortal flesh keep silence, and stand in fear and trembling.

Bearing in mind, of course, that their apophatic point is itself cataphatic. Still. They do have a point.

When we think of doctrine this way, the effect – at least for me, right now – is to cast the *entire history of theological disputation* – and, so, a fortiori, of ecclesial controversy and indeed schism – in a wholly different light than has been our historical wont, in Christendom. Viewed in this light, the controversies just look like *research,* no? Which makes it even more horrible than it already was, that we Christians have been killing each other over it since First Century Alexandria. So stupid.

God grant that the age of that internecine combat among Christians is coming to a close, as our common Enemy makes himself more and more evident in his worldly agents, and they close in upon us, all.

Third, it picks out the limits of natural theology, without at all repudiating it. There are truths that theology can express within and by its propositional system that cannot be derived from it, but *only from revelation.* The Trinity, e.g., cannot be derived from metaphysical First Principles (I think; still working on that), and it cannot be said that we can understand it; but, it can be expressed in the propositional system so far elaborated by orthodox Chalcedonian theology without inconsistency.

Fourth, and perhaps spookiest: the propositional system of Christian theology is capable of accommodating the deliverances of revelation. I am reminded of the spooky way that abstract maths often turn out to inform material reality at the deepest level. Material reality seems to be capable of accommodating abstract mathematics that prima facie seem to have no relation to concrete reals whatever; likewise, the logical calculus of Christian theology seems to be capable of accommodating doctrinal revelations that – to put it in the plainest terms possible – no one left only to his own devices and desires would ever think of.

There was a fifth, but I lost it. Maybe it will come back as comments accrue to this wonderful post.

Ah, this is vintage Kristor! I remember the response by our late great mentor Larry Auster to one of your comments, jocularly lamenting that he was now not educated enough for his own blog because you employed some words he had to look up.

My favorite presuppositional apologist is also the most idiosyncratic: Francis Schaeffer. (Not Francis Schaeffer, Jr, his now-apostate son) Schaeffer

peredid not trumpet his presuppositionalism but instead emphasized the inhumanity of the secular worldview of the post WWII West. Their System did not work, and Schaeffer worked to show that, and to present the life-giving Good News of Jesus Christ.I recommend Schaeffer’s Trilogy:

The God who is there,Escape from Reason, andHe is there and He is not silent.Schaeffer emphasized the unlivability of non-Christian systems; most of the presuppositional establishment emphasizes the logical untenability of basing your thinking on a non-biblical system. This critique is valid in my view, but somewhat elitist.

For more formal, rigorous (“elitist”?) presuppositionalism, Van Til is the best. But my feeling is that rigorous presuppositionalism can be too restrictive if you take it to be a field manual.

Full disclosure: Schaeffer is the only presuppositionalist whose works I have read (almost) fully. I tried Van Til but found him too esoteric.

For a Van-Til-for-Dummies experience, read Greg Bahnsen. The best choice is his

Always Ready: Directions for Defending the Faith*

I’m glad to see that you are stoked, Kristor. When Kristor is stoked, it’s Katy-bar-the-door! In a good way, of course.

*

Permit me to give you a taste of Schaeffer, taken from “The God who is there:”

Alan, my old friend and ally, thank you for your kind words. I reciprocate. I remember that in the old VFR days, when we two were boys pretty much, whenever I saw that Lawrence had posted something from Alan Roebuck, I knew I should put on the old thinking cap, and pay attention.

The quote you offer from Schaeffer pere is an apt synecdoche of the confrontation between apophasy and cataphasy, that must run through all ages. As a mystic, I have lived their controversy in my own being, and it has largely shaped my life, from puberty on. As I descended again from the heights, I knew – as the most fundamental tragedy of life, any life, especially my own – that I could not possibly capture in words or concepts what I had been given to know and understand; and, that it was my duty henceforth to try with all my might to do so.

The bottom line is this: reality as we suffer it from each day to the next, humdrum, worldly, little, paltry, sad, *is and must be throughout and in every detail the work of Infinite Good.* Given the definition of Infinite Good, it could not be otherwise. So, *in its every detail, this world cannot but accommodate Infinite Good.* For, to express that Good even a little is to be an instance of him, complete. This, despite the fact that no lesser thing might possibly comprehend him.

This is how he saves us. When we let him, he magnifies himself in us; so, he magnifies us.

It is why Pelagianism and Gnosticism *cannot work.* Pragmatically, they are massive failures: for, no finite can approach the Throne Room of the Infinite One under his own finite power. To be with the Infinite One, must be the work of the Infinite One.

So anyway, Schaeffer is right. *But so is his Jasperian interlocutor.* On Omniscience, neither perspective can be incorrect. The apophatics and the cataphatics are both right.

So is it that a really deep cataphatic is also an apophatic, and vice versa.

Although Euclid’s parallel postulate may appear intuitively obvious and certainly leads to a self-consistent geometry, so too does Lobachevsky’s alternative: “There exist two lines parallel to a given line through a given point not on the line.” He and Bolyai showed that there can be self-consistent geometries other than Euclid’s.

Euclid stated five postulates on which he based all his theorems:

1) To draw a straight line from any point to any other.

2) To produce a finite straight line continuously in a straight line.

3) To describe a circle with any centre and distance.

4) That all right angles are equal to each other.

5) That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles. (The Parallel Postulate)

Beltrami’s model provides a setting in which Euclid’s first four postulates hold, but the parallel postulate does not. It reduced the problem of consistency of the axioms of non-Euclidean geometry to that of the consistency of the axioms of Euclidean geometry.

As a result, Klein was able to show that there are 3 different types of geometry: In the Bolyai – Lobachevsky type of geometry, straight lines have two infinitely distant points. In the Riemann type of spherical geometry, lines have no (or more precisely two imaginary) infinitely distant points. Euclidean geometry is a limiting case between the two where for each line there are two coincident infinitely distant points.

The extent to which any of these geometries describe the spacio-temporal world we inhabit is not itself a mathematical question

More @ Kristor: The Parallel Postulate is indeed an excellent example of incompleteness. As you may have learned later, other geometries come from varying the postulate: In Lobachevskian Geometry, there are at least two lines through the point and parallel to the line. In elliptic geometery, there are no parallel lines, period.

Euclidean geometry without the parallel postulate is called “natural geometry”. Natural geometry is indeed incomplete (i.e. there are true facts which can be stated in the language of natural geometry but cannot thereby be proved). But with the parallel postulate, Euclidean geometry is, in fact, complete! Crucially, Godel’s first incompleteness theorem only applies to systems capable of discussing the natural numbers and their arithmetic. Euclidean geometry is not capable of doing so, which is why it can be complete. There is no notion of a number (in the pure sense) in language of Euclidean geometry; only points, lines, etc.

I mostly say this as a caution against using the parallel postulate as an example in the context of Godel, because it is really an anti-example, showing where Godel’s first incompleteness theorem does not apply.

Of course, formal philosophical-theological systems are capable of arithmetic (presumably), so Godel would apply, as has been discussed a number of times before here on the Orthosphere. My two cents: I think this is relevant for the Analytic-Continental divide in philosophy: Creating complete, formal analytic philosophical systems is a failed project from the start. Likewise any theology which attempts to rigorously put quasi-mathematical rules around God up to the point of explaining arithmetic (or something equivalent) is a fool’s errand. Doing so, of course, would be to reduce God to a mathematical object, which perhaps would appeal to Spinoza or Liebniz but not to Classical Theists.

Ah, thanks for that reminder, Tenetur. The Parallel Postulate completes Euclidean geometry *without recourse to a more competent logical calculus.*

I’m neither a geometer nor a logician, but granting your point it seems to me nevertheless that, inasmuch as Euclidean geometry can express the Parallel Postulate but cannot demonstrate it, then despite that the First Incompleteness Theorem does not *demonstrate* that Euclidean geometry is incomplete, still it is incomplete: it is completed by propositions that it cannot demonstrate. So, it remains a rather readily comprehensible example of incompleteness for rhetorical purposes, even though it is not covered by the First Incompleteness Theorem. I’m in way over my head, but am I making sense nonetheless?

As for the Analytic/Continental divide, it seems to me that the Continentals threw out the baby with the bath water: since analytical completion is impossible, and consistency is, you know, *hard,* they abandoned formal consistency altogether. So you get pomo insanity, radical incoherence, spiritual despair, and social chaos. I’d rather stick with Anglo-American analytics, while bearing the Gödelian caveat well in mind – which seems an apt counsel of epistemological humility and caution, wholly in keeping with Baconian method.

In the space we inhabit, the Parallel Postulate seems obvious, yet the other axioms cannot prove it. So yes, I would say that Euclidean geometry minus the parallel postulate is incomplete. It misses at least one obvious truth.

The other geometries describe curved spaces, so in them the non-parallel postulates also form incomplete systems which must be completed by a parallel postulate that is appropriate to that particular space.

Hi Kristor,

I would avoid comparing the “incompleteness” of Euclid’s first 4 axioms to “incompleteness” of the Godel type. I believe the latter has to do with propositions that will definitely be true given the axioms of the system, but cannot be formally proved from them. The parallel postulate is more like simple additional, logically independent information.

The interesting part of the story is how mankind was able to imagine for so long that space must necessarily be Euclidean. There must have been a hidden assumption, and it turns out to be the assumption that one knows how to judge line segments separated in space to be parallel. To do this, one must be able to “pick up” a tangent vector from one point and move it while being confident that one is not thereby changing it, and that therefore the path one takes from one segment to the other will not affect the direction of the tangent vector on arrival. This was a natural assumption to make, since the space in which we are embedded has a very nearly Euclidean connection with which to define parallel transport. However, there was this structure there all along, taken for granted and so intimate to our experience that only an act of genius could explicitly recognize it. Even if it were to turn out that our space is exactly the 3-dimensional manifold with Euclidean metric and Euclidean connection structures living on top of it that everyone implicitly assumed it was before Riemann, we would still have a new recognition of these structures as distinct and contingent facts about the world. This seems to me a good model for what we hope to accomplish with metaphysics.

Kristor,

You’re on the right track. If you’re looking for a readily comprehensible example of incompleteness, I would suggest that, rather than Euclidean geometry itself, “natural geometry”, i.e. Euclidean geometry *with the Parallel Postulate removed* is a better example.

Bonald is correct when he says “The parallel postulate is more like simple additional, logically independent information”. The Parallel Postulate (PP) can neither be proved, nor disproved, using the 4 axioms of natural geometry. It can still be expressed though. We can describe the possibility of the PP being true in the language of natural geometry. It’s just that we can’t say anything about its truth, or falsity. In fact, we could, perfectly consistently, assume the PP is false and get non-Euclidean geometries.

But, once you assume the PP as an axiom, you get Euclidean geometry, which *can* (trivially) prove the PP, by assuming it outright as an axiom. It can also prove all other geometric truths expressible in the terms of Euclidean geometry. So Euclidean geometry is complete, while natural geometry is incomplete. Natural geometry *is* a good example of an incomplete formal system, but Euclidean geometry *is complete* since it can prove any statement it can express.

Of course, this technical definition of “prove” is somewhat counter-intuitive: axioms prove themselves! I suppose you could say though that they don’t *justify* themselves. A formal system always needs axioms, but the choice of axioms (and therefore the choice of formal system which you want to study) is contingent. In the language of Alan’s original post, we always must make presuppositions before we can even undergo a formal study of something. Not just the axioms, but the “rules of the game” which let us deduce things from our axioms.

This is not really the point Godel is making though. Godel’s 1st Incompleteness Theorem says that, *even granting the axioms* as perfectly correct, a formal system powerful enough to discuss arithmetic cannot reach a conclusive “verdict” on every statement, i.e. it can’t prove or disprove each and every statement which can be formulated in the language of the formal system. You can keep adding axioms, and it won’t help (up to a point… if you have infinite axioms which are sufficiently complex then the incompleteness theorems break down).

Moreover there will always be *true* statements we cannot prove (in a formal system powerful enough to discuss arithmetic): suppose we have a statement S which we cannot prove or disprove. If it’s true, then S is a true statement we cannot prove. If it’s false, then we can take its negation: “not S” is true and unprovable. So there will always be unprovable, true statements. This is, I think, why Bonald says “I believe [incompleteness in the sense of Godel] has to do with propositions that will definitely be true given the axioms of the system, but cannot be formally proved from them.” But unfortunately I disagree with him that this type of incompleteness is different from the incompleteness of natural geometry.

Bonald:

Yes. The deepest truths and the most constant aspects of experience are the hardest to discern, but also the most fruitful to understand – or, at least, notice, and remember. Also the most important. To take an analogue from social policy: we tend to just assume that civilization will keep replicating itself from one generation to another, even as we delete or deform massive swathes of the procedures that regenerate civilization – or, for that matter, the species.

I’m having trouble sussing the difference between a proposition that is definitely true given the axioms of a formal system – that is, i.e., implicit in them – and a proposition that is entailed by those axioms, so that it can be demonstrated from them. Can you give an example?

Tenetur, thanks again for the clarification: natural geometry is incomplete, while Euclidean geometry – which adds the PP as an axiom to the axioms of natural geometry – is complete.

Does not natural geometry presuppose a flat space in which to define and so scribe a straight line – or, for that matter, to discern whether a line is straight or curved? Does it not, i.e., presuppose the PP, and so implicitly presuppose the axiom set of Euclidean geometry? But the PP is itself expressed in terms of straight lines, so there would seem to be some circularity in the a priori axioms of Euclidean geometry; some codependence or necessary mutual implication, or perhaps then some integrity.

I bet there is a way to use Euclidean geometry to express the propositions of arithmetic.

Kristor,

Regarding your first point: when we discuss geometries as formal systems in logic, we’re not directly studying the geometric objects themselves (points, lines, circles, etc.) nor even their “domain” (the plane, n-dimensional space, etc.). In fact, we are studying a language: expressed using some alphabet of symbols, together with some rules about which symbol combinations are valid (a grammar) and about how to generate new symbol combinations from existing ones. Also, a starting point: the axioms expressed in the language. Now, a formal system for Euclidean geometry gives symbols which we *intend* to represent points, lines, etc. and their relations (intersects, lies-on, etc.). But this interpretation is not built into the structure of the formal language explicitly. The formal language doesn’t “know” anything about flat planes, only about geometries and their relations. And actually not even about those: only about symbols which we intend to represent those, and “the rules of the game”. As far as the system goes, we actually don’t *need* any geometric context whatsoever: so any interpretation you bring is your own. We could even adopt a different interpretation which is basically non-geometric.

Now, in a deeper, less mathematical sense, languages have meaning, and so we can truly say we are discussing geometry when we play with symbols. But fundamentally the rules of the formal system do not depend at all about the context in which lines exist: only their functional relationship with other geometric abstracts. We don’t need to care about whether these lines can actually exist and be constructed and scribed on an actual 2D plane as long as we fully capture the functional behaviour of lines when we make the rules of the formal system. Effectively, we are encoding what it *means* to be a line, in the very most general sense. This, I hope, goes some way toward explaining why the study of natural geometry does not *in itself* assume the PP or the non-curvedness of lines: the rules we have given for lines (or rather, the symbology representing lines in our langauge) simply do not say anything about those things. The language is not equipped to talk about curvedness, and its axioms are not powerful enough to prove PP.

Regarding the second point: Euclidean geometry cannot be interpreted in a way that gives it the power of fully-fledged arithmetic. This follows from the proof of it’s completeness, and the first incompleteness theorem. How could you simulate the multiplication of geometric objects by application of the Euclidean axioms? You could iteratively add… But how many iterations? No matter which interpretative strategy you employ, at some point you need to import further axioms to be able to do arithmetic, at which point you become subject to Godel.

“I’m having trouble sussing the difference between a proposition that is definitely true given the axioms of a formal system – that is, i.e., implicit in them – and a proposition that is entailed by those axioms, so that it can be demonstrated from them.”

This is what’s distinct about Goedel’s (1st) theorem, as I understand it. The theorem has a very specific model of how proofs work in formal systems. It takes there to be a finite list of axioms and rules of inference. Then the question is what statements one can make from the axioms by applying the inference rules on the axioms (and then on combinations of the axioms and the previously thus-established theorems, etc). In principle, this could be done by a computer. The question is, can the computer following this procedure eventually prove or disprove every proposition statable in the system? The proof says “no”. There are some propositions that one can’t “get to” in this way. One way to be in complete, of course, is if the proposition has no definite truth value without further information. (Without the parallel postulate, we don’t have enough information to know if we are in Euclidean or non-Euclidean geometry.) However, the “sentences” that Goedel produced I believe do have definite truth value, but somehow involve self-reference in such a way that they can’t be gotten to by formal inferential manipulation on the axioms. I’m certainly not an expert on this, so I may well be misunderstanding it. If I’m not, then the significance of the theorem is tied to its algorithmic model of theorem-proving, which apparently is not coextensive with mathematical reasoning.

Presuppositionalist apologetics is a truly terrible approach. This article is a good place to start: https://calvinistinternational.com/2014/02/17/covenantal-apologetics-principles-practice-defense-faith/

This video from Lutheran Jordan Cooper is also good: https://youtu.be/LcZqhitfpxo

I have not studied your links yet but I know that presuppositionalism often becomes esoteric and elitist. The presuppositionalist of ill repute rejects the classical proofs of God and of the Bible and insists on one particular approach that appears to the uninitiated to be a form of question-begging. It’s not, but that’s what it looks like to the man in the street.

I am standing on just one part of presuppositional apologetics: everyone has presuppositions which are usually untested and often not consciously known, and these presuppositions determine the outcome of our thought. For this reason, evidence alone is often incorrectly evaluated, and therefore the one who gives a case for Christ must be aware of these difficulties and know how to respond to them.

@ Thursday:

The speaker in the video you linked says that presuppositionalism is a form of anti-realism (idealism), because it holds the knower is primary, rather than reality.

Some presuppositionalists may hold this, but I do not. And that is not what I said in my essay. I said that people draw conclusions based on their presuppositions, and that wrong presuppositions lead to wrong conclusions. I did not say that reality is not primary.

This is the (potential) problem with using labels. People often interpret the label in a way not intended by the author. The phrase “presuppositional apologetics” has different meanings for different people.

Fortunately, I supplied my own meaning of this phrase in the essay. My presuppositionalism means that presuppositions determine thought, and wrong presuppositions lead to wrong thought. By talking of epistemology, I am not saying that it has priority over ontology. It does not.

I couldn’t disagree more. https://philosophical-theology.com

Do you disagree that everyone has presuppositions? Or that presuppositions determine the outcomes of thought?

I disagree presuppositionalism is a terrible approach.

See my reply to “Thursday” above.

I grasp your position. I was responding to another.

Since we are talking about presuppositions, I’ll add that apologetics presupposes that all parties submit to reason. I take apologetics in the narrow sense of an argument that a man can hold the beliefs I do in fact hold, without forfeiting his claim to reason. Christian apologetics is the argument that Christians are not lunatics. In any case, I do not think we can presuppose that all parties nowadays submit to reason, or that there are any arguments, however cogent, that will stop our enemies from locking us up in asylums. I mean, of course, exercising hard discrimination against Christians and counterrevolutionaries.

I noticed that you took the precaution of masking some of your “problematic” phrases from the censor-bots. I do not blame you and may very shortly follow your example. But your understandable use of cryptic typography suggests that we have already moved beyond the world of rational discourse in which apologetics work. Our enemies now “refute” our arguments with censor-bots, deplatforming and the politics of personal destruction. Facebook will not listen to arguments why links this obscure and fusty blog should not be banned from its platform, and we are already responding with a typographic argot like that used by the underworld.

Having gotten that grumpy gloom-and-doom off my chest, I’d like to commend your attention to the epistemic grounds of our own morale. I don’t think we can persuade our enemies that we do not belong in the lunatic asylum, but I do think we must persuade ourselves. As the “social consequences” of faith and dissent become more and more costly, it is more and more important that we believe in what we fight for. This is particularly true of conservatives who do not enjoy nonconformity and being “in tension” with their society.

I am not a logician, and so yield to you and Kristor on questions of systemic rigor, but I do like it when you say, in effect, that the proof is in the pudding. Or as someone somewhere said, “by their fruits you shall know them.” I follow Hegel in the belief that art is the “fruit” by which a worldview shall be known, and say that the harvest of our present worldview is pretty pitiful.

Thanks for your appreciation, JM. Yes, apologetics presupposes that our interlocutor is capable of reasoning about Christianity, and that is less and less common. But it is still sometimes possible. And we have to start by persuading ourselves.

As for masking some of my phrases, remember the full pearls-swine quote: “Give not that which is holy unto the dogs, neither cast ye your pearls before swine, lest they trample them under their feet, and turn again and rend you.” (Matthew 7:6) The Bible itself warns us that swine are dangerous as well as unworthy.

I take the “pearls before swine” passage as one of several Biblical foundations for presuppositionalism. Too many Christians think it is unchristian to say there are “swine” who lack the necessary presuppositions to see “pearls” as “pearls.” I find the same lesson in the passage about “shaking dust from your sandals.” I am sure that grace can transform a “swine” into an aficionado of “pearls,” and likewise transform a house that once repelled an apostle into a house that receives him, but apologetics and evangelism will make no headway if the presuppositions aren’t there. I think this may be what used to be described as the “cast” of a man’s mind. I also think that the “cast” of a mind is as important as, and yet quite different than, its general intelligence. I prefer the company of a simpleton who shares my cast of mind over the company of a brainiac who doesn’t. C.S. Lewis has a good line about this in The Problem of Pain, which I’m afraid I’m right now too lazy to look up.

Talk of presuppositions brought to mind this post on Parmenides: through the law of non-contradiction (the most fundamental of ‘presuppositions’, one might say), Parmenides (or at least the author’s interpretation of Parmenides) provides an absolute refutation of skepticism, and demonstrates that “being is, and cannot not be”:

http://waitingforelijah.blogspot.com/2005_04_01_waitingforelijah_archive.html#111362680458529810

Alan Roebuck makes a very important point: “everyone has presuppositions which are usually untested and often not consciously known, and these presuppositions determine the outcome of our thought. For this reason, evidence alone is often incorrectly evaluated”

Take a simple example. Sometimes, when they can’t find something, one hears people say, “It can’t have simply vanished.” It is something they take for granted.

But how do they know this? It is not a

logicalimpossibility that things should simply vanish; we can quite easily imagine something just disappearing and re-appearing, like the Cheshire Cat in Alice in Wonderland. We know what it would be like for it to be true.This suggests “things don’t simply vanish” is an empirical or scientific proposition, an hypothesis that needs to be tested. But there is an obvious difficulty here; accepting even the possibility of things “simply vanishing” has all sorts of implications for what would

countas testing, proving, how we interpret evidence and our whole system of verification.The solution is simple enough: “things don’t simply vanish” is a rule, a practice, a regulative principle for the way we judge and act, from the most rigorous scientific enquiries to the most ordinary everyday activities. Regarding it as absolutely solid is part of our

methodof doubt and enquiry.Wittgenstein asks us to think of chemical investigations. “Lavoisier makes experiments with substances in his laboratory and now he concludes that this and that takes place when there is burning.

He does not say that it might happen otherwise another time. He has got hold of a definite world-picture – not of course one that he invented: he learned it as a child. I say world-picture and not hypothesis, because it is the matter-of-coursefoundationfor his research and as such also does unmentioned.” (Emphasis added)